The general principles of operation for a plasma centrifuge are well known and well understood. In short, a plasma centrifuge generates forces on charged particles which will cause the particles to separate from each other according to their mass. More specifically, a plasma centrifuge relies on the effect crossed electric and magnetic fields have on charged particles. As is known, crossed electric and magnetic fields will cause charged particles in a plasma to move through the centrifuge on respective helical paths around a centrally oriented longitudinal axis. As the charged particles transit the centrifuge under the influence of these crossed electric and magnetic fields they are, of course, subject to various forces. Specifically, in the radial direction, i.e. a direction perpendicular to the axis of particle rotation in the centrifuge, these forces are: 1) a centrifugal force, F.sub.c, which is caused by the motion of the particle; 2) an electric force, F.sub.E, which is exerted on the particle by the electric field, E.sub.r ; and 3) a magnetic force, F.sub.B, which is exerted on the particle by the magnetic field, B.sub.z. Mathematically, each of these forces are respectively expressed as:
F.sub.c =Mr.omega..sup.2 ; PA1 F.sub.E =eE.sub.r ; and PA1 F.sub.B =er.omega.B.sub.z. PA1 M is the mass of the particle; PA1 r is the distance of the particle from its axis of rotation; PA1 .omega. is the angular frequency of the particle; PA1 e is the electric charge of the particle; PA1 E is the electric field strength; and PA1 B.sub.z is the magnetic flux density of the field. PA1 where we used: EQU .alpha.=(.PHI.-V.sub.ctr)/.PSI.=-2V.sub.ctr /(a.sup.2 B.sub.z) (Eq. 2) PA1 and where a is the radius of the chamber.
Where:
In a plasma centrifuge, it is general practice that the electric field will be directed radially inward. Stated differently, there is an increase in positive voltage with increased distance from the axis of rotation in the centrifuge. Under these conditions, the electric force F.sub.E will oppose the centrifugal force F.sub.C acting on the particle, and depending on the direction of rotation, the magnetic force either opposes or aids the outward centrifugal force. Accordingly, an equilibrium condition in a radial direction of the centrifuge can be expressed as: EQU .SIGMA.F.sub.r =0 (positive direction radially outward) EQU F.sub.c -F.sub.E -F.sub.B =0 EQU Mr.omega..sup.2 -eE.sub.r -er.omega.B.sub.z =0 (Eq. 1)
It is noted that Eq. 1 has two real solutions, one positive and one negative, namely: EQU .omega.=.OMEGA./2(1.+-.1+4E.sub.r +L /(rB.sub.z +L .OMEGA.))
where .OMEGA.=eB.sub.z /M.
For a plasma centrifuge, the intent is to seek an equilibrium to create conditions in the centrifuge which allow the centrifugal forces, F.sub.c, to separate the particles from each other according to their mass. This happens because the centrifugal forces differ from particle to particle, according to the mass (M) of the particular particle. Thus, particles of heavier mass experience greater F.sub.c and move more toward the outside edge of the centrifuge than do the lighter mass particles which experience smaller centrifugal forces. The result is a distribution of lighter to heavier particles in a direction outward from the mutual axis of rotation. As is well known, however, the separation achieved in a plasma centrifuge is limited.
As an alternative to the plasma centrifuge, an apparatus which is structurally similar but which is operationally and functionally very dissimilar has been more recently developed. This alternative apparatus is referred to herein as a plasma mass filter and is fully disclosed in co-pending U.S. application Ser. No. 09/192,945 for an invention of Ohkawa entitled "Plasma Mass Filter" which is assigned to the same assignee as the present invention. The fundamental difference between a plasma centrifuge and a plasma mass filter is that, unlike a plasma centrifuge which relies on collisions between the various ions as they are rotated in the plasma chamber, a plasma mass filter relies on the ability of the ions to orbit inside the plasma chamber. Thus, the physics that are involved are different.
As indicated above in connection with Eq. 1, a force balance can be achieved for all conditions when the electric field E is chosen to confine ions, and ions exhibit confined orbits. In a plasma filter, however, unlike a centrifuge, the electric field is chosen with the opposite sign to extract ions. The result is that ions of mass greater than a cut-off value, M.sub.c, are on unconfined orbits. The cut-off mass, M.sub.c, can be selected by adjusting the strength of the electric and magnetic fields. The basic features of the plasma filter can be described using the Hamiltonian formalism.
The total energy (potential plus kinetic) is a constant of the motion and is expressed by the Hamiltonian operator: EQU H=e.PHI.+(P.sub.R.sup.2 +P.sub.z.sup.2)/(2M)+(P.sub..theta. -e.PSI.).sup.2 /(2Mr.sup.2)
where P.sub.R =MV.sub.R, P.sub..theta. =MrV.sub..theta. +e.PSI., and P.sub.z =MV.sub.z are the respective components of the momentum and e.PHI. is the potential energy. .PSI.=r.sup.2 B.sub.z /2 is related to the magnetic flux function and .PHI.=.alpha..PSI.+V.sub.ctr is the electric potential. E=-.gradient..PHI. is the electric field which is chosen to be greater than zero for the filter case of interest. We can rewrite the Hamiltonian: EQU H=e.alpha.r.sup.2 B.sub.z /2+eV.sub.ctr +(P.sub.R.sup.2 +P.sub.z.sup.2)/(2M)+(P.sub..theta. -er.sup.2 B.sub.z /2).sup.2 /(2Mr.sup.2)
We assume that the parameters are not changing along the z axis, so both P.sub.z and P.sub..theta. are constants of the motion. Expanding and regrouping to put all of the constant terms on the left hand side gives: EQU H-eV.sub.ctr -P.sub.z.sup.2 /(2M)+(P.sub..theta..OMEGA./2=P.sub.R.sup.2 /(2M)+(P.sub..theta..sup.2 /(2Mr.sup.2)+(M.OMEGA.r.sup.2 /2)(.OMEGA./4+.alpha.)
where .OMEGA.=eB/M.
The last term is proportional to r.sup.2, so if .OMEGA./4+.alpha.&lt;0 then, since the second term decreases as 1/r.sup.2, P.sub.R.sup.2 must increase to keep the left-hand side constant as the particle moves out in radius. This leads to unconfined orbits for masses greater than the cut-off mass given by: EQU M.sub.c =e(B.sub.2 a).sup.2 /(8V.sub.ctr)
So, for example, normalizing to the proton mass, M.sub.p, we can rewrite Eq. 2 to give the voltage required to put higher masses on loss orbits: EQU V.sub.ctr &gt;1.2.times.10.sup.-1 (a(m)B(gauss)).sup.2 /(M.sub.C /M.sub.P)
Hence, a device radius of 1 m, a cutoff mass ratio of 100, and a magnetic field of 200 gauss require a voltage of 48 volts.
The same result for the cut-off mass can be obtained by looking at the simple force balance equation given by: EQU .SIGMA.F.sub.r =0 (positive direction radially outward) EQU F.sub.c +F.sub.E +F.sub.B =0 EQU Mr.omega..sup.2 +eEr-er.omega.B.sub.z =0 (Eq. 3)
which differs from Eq. 1 only by the sign of the electric field and has the solutions: EQU .omega.=.OMEGA./2(1.+-.1-4E+L /(rB.sub.z +L .OMEGA.))
so if 4E/rB.sub.z.OMEGA.&gt;1 then .omega. has imaginary roots and the force balance cannot be achieved. For a filter device with a cylinder radius "a", a centralvoltage, V.sub.ctr, and zero voltage on the wall, the same expression for the cut-off mass is found to be: EQU M.sub.C =ea.sup.2 B.sub.Z.sup.2 /8V.sub.ctr (Eq. 4)
When the mass M of a charged particle is greater than the threshold value (M&gt;M.sub.c), the particle will continue to move radially outwardly until it strikes the wall, whereas the lighter mass particles will be contained. The higher mass particles can also be recovered from the walls using various approaches.
It is important to note that for a given device the value for M.sub.c in equation 3 is determined by the magnitude of the magnetic field, B.sub.Z, and the voltage at the center of the chamber (i.e. along the longitudinal axis), V.sub.ctr. These two variables are design considerations and can be controlled. It is also important that the filtering conditions (Eqs. 2 and 3) are not dependent on boundary conditions. Specifically, the velocity and location where each particle of a multi-species plasma enters the chamber does not affect the ability of the crossed electric and magnetic fields to eject high-mass particles (M&gt;M.sub.c) while confining low-mass particles (M&lt;M.sub.c) to orbits which remain within the distance "a" from the axis of rotation.
It happens that in a plasma mass filter, wherein ions are subjected to the conditions disclosed above, those ions which have a mass greater than the cut-off value, M.sub.c, will follow unconfined orbits that cause them to be rapidly ejected from the space where ions having a mass less than the cut-off value are confined. Actually, this separation typically occurs in less than one-half of a rotation of a multi-species plasma about its axis of rotation. Due to this quite rapid separation of heavy mass particles from light mass particles, the present invention recognizes that it is not necessary for the multi-species plasma to be moved in translation through the plasma chamber. Instead, the particles can be separated in the plasma according to their mass while being constrained to move primarily in rotation.
In light of the above, it is an object of the present invention to provide an enclosed toroidal plasma mass filter having a partially-cylindrical plasma chamber wherein the source of a multi-species plasma is azimuthally distanced from the collector that is to be used for collecting the light mass ions from the plasma, while the heavy mass ions are ejected into the chamber wall. It is another object of the present invention to provide a toroidal plasma mass filter which is relatively compact and efficiently conserves space. Yet another object of the present invention is to provide a toroidal plasma mass filter wherein the crossed electric and magnetic fields in the plasma chamber act to draw the multi-species plasma from its source into the chamber. Still another object of the present invention is to provide a toroidal plasma mass filter wherein antennae can be located sufficiently near the source of the multi-species plasma to heat electrons at the source. Another object of the present invention is to provide a toroidal plasma mass filter in which the magnetic field is oriented in the plasma chamber so that electrical disturbances at the ion collector are impeded from propagating back upstream to the source in a direction perpendicular to the magnetic field. Another object of the present invention is to provide a toroidal plasma mass filter which is relatively easy to manufacture, functionally simple to operate, and comparatively cost effective.